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Mirrors > Home > MPE Home > Th. List > dftpos2 | Structured version Visualization version GIF version |
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dftpos2 | ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmtpos 7646 | . . 3 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | |
2 | 1 | reseq2d 5642 | . 2 ⊢ (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹 ↾ ◡dom 𝐹)) |
3 | reltpos 7639 | . . 3 ⊢ Rel tpos 𝐹 | |
4 | resdm 5691 | . . 3 ⊢ (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹 |
6 | df-tpos 7634 | . . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
7 | 6 | reseq1i 5638 | . . 3 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) |
8 | resco 5893 | . . 3 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) | |
9 | ssun1 3998 | . . . . 5 ⊢ ◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅}) | |
10 | resmpt 5699 | . . . . 5 ⊢ (◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹) = (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹) = (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) |
12 | 11 | coeq2i 5528 | . . 3 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) |
13 | 7, 8, 12 | 3eqtri 2805 | . 2 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) |
14 | 2, 5, 13 | 3eqtr3g 2836 | 1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∪ cun 3789 ⊆ wss 3791 ∅c0 4140 {csn 4397 ∪ cuni 4671 ↦ cmpt 4965 ◡ccnv 5354 dom cdm 5355 ↾ cres 5357 ∘ ccom 5359 Rel wrel 5360 tpos ctpos 7633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-tpos 7634 |
This theorem is referenced by: tposf12 7659 |
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